The strain tensor, ε, is a symmetric tensor used to quantify the strain of an object undergoing a small 3-dimensional deformation:

• the diagonal coefficients ε_ii are the relative change in length in the direction of the i direction (along the x_i-axis) ;
• the other terms ε_ij = 1/2 γ_ij (i ≠ j) are the shear strains, i.e. half the variation of the right angle (assuming a small cube of matter before deformation).

The deformation of an object is defined by a tensor field, i.e., this strain tensor is defined for every point of the object. This field is linked to the field of the stress tensor by the generalized Hooke's law.

In case of small deformations, the strain tensor is the Green tensor or Cauchy's infinitesimal strain tensor, defined by the equation:

$\backslash varepsilon\_\{ij\}\; =\; \{1\; \backslash over\; 2\}\; \backslash left\; (\{\backslash part\; u\_i\; \backslash over\; \backslash part\; x\_j\}\; +\; \{\backslash part\; u\_j\; \backslash over\; \backslash part\; x\_i\}\backslash right\; )$

Where u represents the displacement field of the object's configuration (i.e., the difference between the object's configuration and its natural state). This is the 'symmetric part' of the Jacobian matrix.

For large (finite) deformations see Finite Deformation Tensors.

Demonstration in simple cases

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One-dimensional elongation

When the segment, parallel to the x₁-axis, is deformed to become the [A'B' segment, the deformation being also parallel to x₁

the ε₁₁ strain is (expressed in algebraic length):

$\backslash varepsilon\_\{11\}\; =\; \backslash frac\{\backslash Delta\; l\}\{l\_0\}\; =\; \backslash frac\{\backslash overline\{A\text{'}B\text{'}\}-\backslash overline\{AB\}\}\{\backslash overline\{AB\}\}$

Considering that

$\backslash overline\{AA\text{'}\}\; =\; u\_1(A)$ and $\backslash overline\{BB\text{'}\}\; =\; u\_1(B)$

the strain is

$\backslash varepsilon\_\{11\}\; =\; \backslash frac\{\backslash overline\{AB\}\; +\; \backslash overline\{BB\text{'}\}-\backslash overline\{A\text{'}A\}\}\{\backslash overline\{AB\}\}\; -\; 1$

$\backslash varepsilon\_\{11\}\; =\; \backslash frac\{u\_1(B)-u\_1(A)\; +\; \backslash overline\{AB\}\}\{\backslash overline\{AB\}\}\; -\; 1$

The series expansion of u₁ is

$u\_1(B)\; \backslash simeq\; u\_1(A)\; +\; \backslash frac\{\backslash partial\; u\_1\}\{\backslash partial\; x\_1\}\; \backslash cdot\; \backslash overline\{AB\}$

and thus

$\backslash varepsilon\_\{11\}\; =\; \backslash frac\{\backslash partial\; u\_1\}\{\backslash partial\; x\_1\}$

And in general

$\backslash varepsilon\_\{ii\}\; =\; \backslash frac\{\backslash partial\; u\_i\}\{\backslash partial\; x\_i\}\; =\; \backslash frac\{1\}\{2\}\; \backslash left\; (\; \backslash frac\{\backslash partial\; u\_i\}\{\backslash partial\; x\_i\}\; +\; \backslash frac\{\backslash partial\; u\_i\}\{\backslash partial\; x\_i\}\; \backslash right\; )$

Pure shear strain

Let us now consider a pure shear strain. An ABCD square, where is parallel to x₁ and [AD is parallel to x₂, is transformed into a AB'C'D' rhombus, symmetric to the first bisecting line.

The tangent of the γ angle is:

$\backslash tan(\backslash gamma)\; =\; \backslash frac\{\backslash overline\{BB\text{'}\}\}\{\backslash overline\{AB\}\}$

for small deformations,

$\backslash tan(\backslash gamma)\; \backslash simeq\; \backslash gamma$

and

$\backslash overline\{BB\text{'}\}\; =\; u\_2(B)\; \backslash simeq\; u\_2(A)\; +\; \backslash frac\{\backslash partial\; u\_2\}\{\backslash partial\; x\_1\}\; \backslash cdot\; \backslash overline\{AB\}$

and u₂(A) = 0. Thus,

$\backslash gamma\; \backslash simeq\; \backslash frac\{\backslash partial\; u\_2\}\{\backslash partial\; x\_1\}$

Considering now the ^* segment:

$\backslash gamma\; \backslash simeq\; \backslash frac\{\backslash partial\; u\_1\}\{\backslash partial\; x\_2\}$

and thus

$\backslash gamma\; =\; \backslash frac\{1\}\{2\}\; \backslash gamma\_\{12\}\; =\; \backslash varepsilon\_\{12\}\; =\; \backslash frac\{1\}\{2\}\; \backslash left\; (\; \backslash frac\{\backslash partial\; u\_1\}\{\backslash partial\; x\_2\}\; +\; \backslash frac\{\backslash partial\; u\_2\}\{\backslash partial\; x\_1\}\; \backslash right\; )$

where γ₁₂ is the engineering strain, which is equal to 2γ.

It is interesting to use the average because the formula is still valid when the rhombus rotates; in such a case, there are two different angles $\backslash gamma\_B\; =\; \backslash widehat\{B\text{'}AB\}$ et $\backslash gamma\_D\; =\; \backslash widehat\{D\text{'}AD\}$ and the formula allows for neglecting the variation of angle due to rigid-body motion (which gives no contribution to the strain).

Relative variation of the volume

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The dilitation (the relative variation of the volume) δ = ΔV/V₀, is the trace of the tensor:

$\backslash delta=\backslash frac\{\backslash Delta\; V\}\{V\_0\}\; =\; \backslash varepsilon\_\{11\}\; +\; \backslash varepsilon\_\{22\}\; +\; \backslash varepsilon\_\{33\}$

Actually, if we consider a cube with an edge length a, it is a quasi-cube after the deformation (the variations of the angles do not change the volume) with the dimensions $a\; \backslash cdot\; (1\; +\; \backslash varepsilon\_\{11\})\; \backslash times\; a\; \backslash cdot\; (1\; +\; \backslash varepsilon\_\{22\})\; \backslash times\; a\; \backslash cdot\; (1\; +\; \backslash varepsilon\_\{33\})$ and V₀ = a³, thus

$\backslash frac\{\backslash Delta\; V\}\{V\_0\}\; =\; \backslash frac\{\backslash left\; (\; 1\; +\; \backslash varepsilon\_\{11\}\; +\; \backslash varepsilon\_\{22\}\; +\; \backslash varepsilon\_\{33\}\; +\; \backslash varepsilon\_\{11\}\; \backslash cdot\; \backslash varepsilon\_\{22\}\; +\; \backslash varepsilon\_\{11\}\; \backslash cdot\; \backslash varepsilon\_\{33\}+\; \backslash varepsilon\_\{22\}\; \backslash cdot\; \backslash varepsilon\_\{33\}\; +\; \backslash varepsilon\_\{11\}\; \backslash cdot\; \backslash varepsilon\_\{22\}\; \backslash cdot\; \backslash varepsilon\_\{33\}\; \backslash right\; )\; \backslash cdot\; a^3\; -\; a^3\}\{a^3\}$

as we consider small deformations,

$1\; \backslash gg\; \backslash varepsilon\_\{ii\}\; \backslash gg\; \backslash varepsilon\_\{ii\}\; \backslash cdot\; \backslash varepsilon\_\{jj\}\; \backslash gg\; \backslash varepsilon\_\{11\}\; \backslash cdot\; \backslash varepsilon\_\{22\}\; \backslash cdot\; \backslash varepsilon\_\{33\}$

therefore the formula.

Real variation of volume (top) and the approximated one (bottom): the green drawing shows the estimated volume and the orange drawing the neglected volume

In case of pure shear, we can see that there is no change of the volume.

Derivation of the strain tensor

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Let the position of a point in a material be specified by a vector $\backslash mathbf\{x\}$ with components $x\_i$. Let the point then move a small distance to a new position specified by a vector with components

$x\text{'}\_i=x\_i+u\_i(\backslash mathbf\{x\})\backslash ,$

where $u\_i$ is a vector function of $\backslash mathbf\{x\}$. Let $x\_i+dx\_i$ be a point close to $x\_i$. After the motion, it will be in a new position given by:

$x\text{'}\_i+dx\text{'}\_i\; =\; x\_i+dx\_i\; +\; u\_i(\backslash mathbf\{x\}\backslash !+\backslash !\backslash mathbf\{dx\})\backslash ,$

Since the $u\_i$ are small, we may approximate them by the first two terms in their Taylor series

$x\text{'}\_i+dx\text{'}\_i\; \backslash approx\; x\_i+dx\_i\; +\; u\_i\; +\; (\backslash partial\_j\; u\_i)\backslash ,dx\_j$

where we have used $\backslash partial\_j$ to represent $\backslash partial/\backslash partial\; x\_j$ and we have used Einstein notation in which repeated indices in a product are assumed to be summed (i.e. index j in this case). $\backslash partial\_j\; u\_i$ is just the Jacobian matrix of the $u\_i$ function. If we represent the unit matrix by $\backslash delta\_\{ij\}$ then the above equation may be written:

$x\text{'}\_i+dx\text{'}\_i\; =\; x\text{'}\_i\; +\; (\backslash delta\_\{ij\}+\backslash partial\_j\; u\_i)\backslash ,dx\_j$

It is seen that the final term (the displacement matrix) specifies the infinitesimal change in the position ($dx\text{'}\_i$) of the nearby particle. If the $u\_i$ are constants, the displacement matrix will be the unit matrix, and the resulting displacement will simply be a rigid translation. Any matrix may be written as the sum of an antisymmetric matrix and a symmetric matrix. Writing the diplacement matrix (in parentheses in the above equation) in this manner yields:

$\backslash delta\_\{ij\}+\backslash frac\{1\}\{2\}(\backslash partial\_j\; u\_i-\backslash partial\_i\; u\_j)\; +\backslash frac\{1\}\{2\}(\backslash partial\_j\; u\_i+\backslash partial\_i\; u\_j)$

The first two terms are the unit matrix and the antisymmetric part of the displacement matrix. These are the first two terms in the Taylor series of an infinitesimal rigid rotation about the translated point $x\text{'}\_i$, and therefore do not represent a deformation of the material. It is the second, symmetric matrix which represents the deformation of the material and this is just the strain tensor $\backslash varepsilon\_\{ij\}$:

$\backslash varepsilon\_\{ij\}=\backslash frac\{1\}\{2\}(\backslash partial\_j\; u\_i+\backslash partial\_i\; u\_j)$

References

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continuum mechanicsTensors | Verzerrungstensor | Tensor tensión | Tenseur des déformations